Fields, noise, and flow
Pink Noise
A scale-balanced fluctuation with power inversely proportional to frequency.
What it is
Pink noise, also called 1/f noise, is defined by a power spectral density S(f) proportional to 1/f over a stated frequency range. Each octave then contributes comparable power: the band from 1 to 2 Hz carries the same total power as 2 to 4 Hz. [1]
It is not “a pink-looking random image.” It is a statement about how variance is distributed across scales. Compared with white noise’s equal power per unit frequency, pink noise gives greater amplitude to slower change while still retaining detail at faster scales.
Mathematical construction
A finite signal can be constructed in the frequency domain: draw complex white-noise coefficients, multiply their amplitude by 1/√f (so power falls as 1/f), impose conjugate symmetry for a real signal, then apply an inverse Fourier transform. For a general coloured-noise exponent α, S(f) ∝ 1/fα; pink is α = 1, white is α = 0, and Brownian/red noise is commonly α = 2.
Equivalent approximations use a bank of random processes updated at different rates, or a filter applied to white noise. Any finite implementation needs lower and upper frequency limits: the ideal 1/f expression cannot extend indefinitely to zero frequency with finite total power.
Reading the spectrum
On logarithmic axes, A flat white-noise spectrum appears horizontal; the pink spectrum drops one power decade for each frequency decade. Its visual consequence is nested variation: slow drifts establish a large structure while smaller fluctuations remain present. This is a spectral relationship, not the spatial minimum-distance condition of blue noise and not the gradient interpolation construction of Perlin noise.
Structural use
- Multi-scale variation. Distribute change across coarse and fine levels without making every level equally dominant.
- Temporal modulation. Create a parameter that drifts with memory rather than jumping independently frame to frame.
- Hierarchy. Vary spacing, density, displacement, or pressure with broad organisation and smaller local disturbance.
- Analysis. Compare procedural output by its spectral slope instead of relying only on visual judgement.
Algorithmics note. Pink noise is best treated as a frequency-distribution tool. It can modulate a field or a parameter over time or space; it does not itself specify a point sampler or a geometry.